A national study reports that 20% of college freshman have GPA above 3.5. We wish to determine if the corresponding proportion is higher at a local university. To conduct the test we select a random sample of 100 freshmen and determine that 26 of them have GPA above 3.5. Run the test at a 5% significance using the p – value approach.
To test the hypothesis, we will follow the p-value approach at a 5% significance level.
Step 1: Define the null and alternative hypotheses.
The null hypothesis (H0) assumes that the proportion of college freshmen with GPA above 3.5 is the same at the local university as the national average of 20%.
H0: p = 0.20
The alternative hypothesis (Ha) assumes that the proportion of college freshmen with GPA above 3.5 is higher at the local university.
Ha: p > 0.20
Step 2: Calculate the test statistic.
To calculate the test statistic, we need to compute the z-score using the sample proportions.
Sample proportion, p̂ = x / n
where x is the number of freshmen with GPA above 3.5 in the sample, and n is the sample size.
p̂ = 26 / 100 = 0.26
Under the null hypothesis, the proportion is assumed to be equal to the national average of 20%, which is p = 0.20.
The standard deviation (σ) for the sampling distribution of proportions can be calculated using the formula:
σ = sqrt(p * (1 - p) / n)
where n is the sample size.
σ = sqrt(0.20 * (1 - 0.20) / 100) = 0.039597
Now, the z-score can be calculated using the formula:
z = (p̂ - p) / σ
z = (0.26 - 0.20) / 0.039597 = 1.5108 (rounded to four decimal places)
Step 3: Calculate the p-value.
The p-value is the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. Since we have a one-tailed test (Ha: p > 0.20), we need to find the area under the standard normal distribution curve to the right of the calculated z-score.
To find the p-value, we can use a z-table or an online calculator. The table or calculator will provide the probability associated with the z-score of 1.5108, which corresponds to 0.9357 (rounded to four decimal places).
However, since we are conducting a one-tailed test, we need to find the area to the right of the z-score. The p-value can be calculated by subtracting this area from 1.
p-value = 1 - 0.9357 = 0.0643 (rounded to four decimal places)
Step 4: Make a decision.
Compare the p-value (0.0643) to the significance level (0.05) to make a decision.
If the p-value is less than or equal to the significance level (p ≤ α), then we reject the null hypothesis in favor of the alternative hypothesis. In this case, the p-value is greater than the significance level (0.0643 > 0.05).
Therefore, we do not have enough statistical evidence to reject the null hypothesis. There is insufficient evidence to suggest that the proportion of college freshmen with GPA above 3.5 is higher at the local university compared to the national average.